FACULTY OF SCIENCE UNIVERSITY OF COPENHAGEN

Master Project in Mathematics Paolo Masulli

Stable homotopy and the Adams Spectral Sequence

Advisor: Jesper Grodal

Handed-in: January 24, 2011 / Revised: March 20, 2011 CONTENTS i

Abstract In the first part of the project, we define stable homotopy and construct the category of CW-spectra, which is the appropriate setting for studying it. CW-spectra are particular spectra with properties that resemble CW-complexes. We define the homotopy groups of CW- spectra, which are connected to stable homotopy groups of spaces, and we define homology and cohomology for CW-spectra. Then we construct the Adams spectral sequence, in the setting of the category of CW-spectra. This spectral sequence can be used to get information about the stable homotopy groups of spaces. The last part of the project is devoted to calculations. We show two examples of the use of the Adams sequence, the first one applied to the stable homotopy groups of spheres and the second one to the homotopy of the spectrum ko of the connective real K-theory.

Contents

Introduction 1

1 Stable homotopy 2 1.1 Generalities ...... 2 1.2 Stable homotopy groups ...... 2

2 Spectra 3 2.1 Definition and homotopy groups ...... 3 2.2 CW-Spectra ...... 4 2.3 Exact sequences for a cofibration of CW-spectra ...... 11 2.4 Cohomology for CW-spectra ...... 13

3 The Adams Spectral Sequence 17 3.1 Exact resolutions ...... 17 3.2 Constructing the spectral sequence ...... 18 3.3 The main theorem ...... 20

4 Calculations 25 4.1 Stable homotopy groups of spheres ...... 25 4.2 The homotopy of the spectrum ko ...... 28

References 34 CONTENTS 1

Introduction

The stable homotopy groups of spaces are quite complicated to compute and this spectral sequence provides a tool that helps in this task. To get a clean construction of the spectral sequence, instead of working with space, we consider spectra, which are sequences of spaces (Xi) with structure maps ΣXi → Xi+1 and soon we restrict our attention to CW-spectra. So we construct a suitable category for these objects, in which many of the results that hold for CW-complexes are true. The construction of the spectral sequence is done following the approach of [Hat04]. Performing calculations of stable homotopy groups using of the Adams spectral sequence is not a mechanical process. One has to compute free resolutions over the Steenrod Algebra and the differentials of the spectral sequence to obtain the E∞ page. Moreover, the E∞ term gives only sub- quotients of the p-components of the stable homotopy groups for certain filtrations of them. So one has still to solve the extension problem to get the group itself. In Section 4, we show two examples of this process: the first one regards the stable homotopy groups of spheres, the second one the homotopy of the spectrum ko of the connective real K-theory. We work out the cited free resolutions explicitly, by actually computing the generators of each free module. We assume that the reader is familiar with the material contained in [Hat02]. The proofs of some of the results here presented could not fit into this project: in such cases we sketch them and refer the reader to one of the references for a more detailed treatment.

Acknowledgements I am grateful to all the people I have interacted with, during the last two years at the Department of Mathematical Sciences at the University of Copenhagen. A special thanks goes to Samik Basu, for suggesting me an alternative way to perform the computations of exact resolutions over A(1) and for his willingness to explain that to me. I am especially grateful to my advisor, Professor Jesper Grodal. He pro- posed me a challenging topic for this project and furthermore he suggested to look at the computations for the spectrum ko, which turned out to be very interesting. I am in debt with him for his support and help every time I needed advices or clarifications. 2

1 Stable homotopy

1.1 Generalities In this section, we will always consider basepointed topological spaces with a CW-structure, i.e. pairs (X, x0) where the space X is a CW-complex and the basepoint x0 ∈ X is a fixed 0-cell.

Definition 1.1. Given two basepointed spaces (X, x0) and (Y, y0), their smash product is defined as

X ∧ Y = (X × Y )/(X ∨ Y ) = (X × Y )/(X × {y0} ∪ {x0} × Y ). This is again a basepointed space, with basepoint the subspace we quotient out. For a basepointed space X, we consider (instead of the suspension) the reduced suspension, which is

ΣX := (SX)/({x0} × I), where SX is the suspension of X. The reduced suspension is again a base- pointed space and it follows immediately for the definition that it can be ex- pressed in terms of smash product with the sphere S1, namely ΣX = X ∧S1. Let us also note that, if X is a CW-complex, the reduced suspension is homotopy equivalent to the unreduced one. In fact ΣX is obtained from SX by collapsing the contractible subspace {x0} × I.

1.2 Stable homotopy groups The term stable in Algebraic Topology is used to indicate phenomena that occur in any dimension, without depending essentially on it, given that the dimension is sufficiently large. [Ada74, Part III, Ch. 1] Let us recall a classical result: Theorem 1.2. (Freudenthal Theorem) Let X be a (n − 1)-connected CW- complex. Then the suspension map πi(X) → πi+1(ΣX) is an isomorphism for i < 2n − 1 and a surjection for i = 2n − 1. If X is n-connected, then ΣX is (n+1)-connected and so the Freudenthal theorem implies that, if X is a CW-complex of any connection and i ∈ N, the morphisms in the sequence:

k πi(X) → πi+1(ΣX) → · · · → πi+k(Σ X) → ... will become isomorphisms from a certain point. The group to which the sequence stabilized is called the ith stable homotopy group of X, denoted s πi (X). This is the same as saying that: πs(X) = lim π (ΣkX), i −→ i+k 3 where lim denotes the direct limit. −→ 0 s 0 th In particular, taking X = S , πi (S ) is called the i stable homotopy s groups of spheres and is denoted by πi . By the Freudenthal Theorem, we have: s s 0 ∼ n πi = πi (S ) = πn+i(S ), for any n > i + 1. The composition of maps Si+j+k → Sj+k → Sk induces a product struc- s L s ture on π∗ = i πi . We have the following: s s s Proposition 1.3. The composition product πi × πj → πi+j given by com- s position makes π∗ into a graded ring in which we have the relation αβ = ij s s (−1) βα, for α ∈ πi , β ∈ πj . Proof. See [Hat02, Proposition 4.56].

2 Spectra

2.1 Definition and homotopy groups

Definition 2.1. A spectrum E is a family (En)n∈N of basepointed spaces Xn together with structure maps εn :ΣEn → En+1 that are basepoint- preserving. n Example 2.2. Let X be a pointed space. If we set En = Σ (X) for all n+1 n ∈ N and εn the identity of Σ we get a spectrum, called the suspension spectrum of X. In particular, taking X = S0, we get the sphere spectrum, denoted S0. The nth space of this spectrum is homeomorphic to Sn. A second example shows the connection of spectra with cohomology: by the Brown Representation Theorem [Hat02, Thm 4E.1], for every reduced cohomology theory K˜ on the category of CW-complexes with basepoint, n we have that the functor K˜ is natural equivalent to [−,En] for a family of basepointed CW-complexes (En), where [X,En] denotes the set of the homotopy classes of basepoint-preserving maps X → En. By the suspension isomorphism for K, we have that: σ : K˜ n(X) → K˜ n+1(ΣX) is a natural isomorphism. Hence we get the sequence of natural isomorphisms: ∼ n ∼ n+1 ∼ ∼ [X,En] = K˜ (X) = K˜ (ΣX) = [ΣX,En+1] = [X, ΩEn+1], where the last equivalence comes from the adjunction between reduced sus- pension and basepointed loops. By the (contravariant) Yoneda Lemma, the ∼ natural equivalence [X,En] = [X, ΩEn+1] must be induced by a weak equiv- 0 alence εn : En → ΩEn+1 which is the adjoint of a map εn :ΣEn → En+1 and so (En) forms a spectrum. If we take the reduced cohomology with coefficients in an abelian group G, we get En = K(G, n) and the spectrum is called the Eilenberg-MacLane spectrum for the group G and denoted HG. 2.2 CW-Spectra 4

This can be slightly generalized by shifting dimension of an integer m and taking Xm = K(G, n + m) with the same maps ΣK(G, n + m) → K(G, n + m + 1). We will denote this shifted Eilenberg-MacLane with K(G, n). It is possible to define homotopy groups for spectra in a quite natural way: Definition 2.3. Let E be a spectrum. We define the ith homotopy group of E, denoted πi(E), as the direct limit of the sequence:

Σ πi+n+1(εn) · · · → πi+n(En) −→ πi+n+1(ΣEn) −−−−−−−→ Σ πi+n+1(En+1) −→ πi+n+2(ΣEn+1) → ....

In symbols: π (E) = lim π (E ). i −→ i+n n Example 2.4. The homotopy groups of the suspension spectrum of a space are precisely the stable homotopy groups of that space, since the structure maps εi are all identities. For the Eilenberg-MacLane spectrum, we already know that ΣEn → En+1 is a weak equivalence for all n and the homomor- phism πr+n+i(En+1) → πr+n+i+1(ΣEn+i) is an isomorphism in the range given by the Freudenthal suspension theorem, so we have that, as for the Eilenberg-MacLane spaces: ( G, if i = 0, πi(HG) = 0, if i 6= 0.

Note that, unlike in the definition of the stable homotopy group of a space, this direct limit is not always attained. Let us give the definition of a subobject of a spectrum, which will be useful for the next section: Definition 2.5. Given a spectrum E, a subspectrum A of E is a family of spaces (An)n∈N such that An ⊆ En and εn(ΣAn) ∈ An+1 for all n ∈ N. In particular, A is a spectrum if we take εn|ΣAn as structure maps.

2.2 CW-Spectra Our aim now is to build a category with spectra as objects and use this setting to study stable homotopy phenomena. To do this we will consider a particular class of spectra, called CW-spectra, which have a structure that resembles the one of CW-complexes. Given a spectrum E in which the spaces En are a CW-complexes with a 0-cell as basepoint for all n ∈ N, we can apply the procedure that we sketch here: first we make the structure maps cellular, via the cellular approxi- mation theorem ([Hat02, Thm. 4.8]): since the structure maps of E map 2.2 CW-Spectra 5 basepoints to basepoints and they are 0-cells in the CW-structures, we can take the cellular maps to be also basepoint-preserving. Then we proceed replacing each space En with union of the reduced mapping cylinders of the maps in the composition1, which is again a CW-complex:

n−1 n Σ ε0 n−1 εn−1 Σ (E0) −−−−−→ Σ (E1) → · · · → Σ(En−1) −−−→ En.

For example, we begin the construction by replacing E1 with the homotopy- 0 equivalent space E1 = Mε0 , so that we have an inclusion ΣE0 ,→ Mε0 . Then we consider 2 Σε0 ε1 Σ (E0) −−→ Σ(E1) −→ E2 0 0 and we have that ΣE1 = MΣε0 ,→ MΣε0 ∪ Mε1 ' Mε1◦Σε0 = E2 ' E2. Proceeding this way we get a new spectrum in which all spaces are CW- complexes with the basepoints as 0-cells and the structure maps are in- clusions of subcomplexes. This construction gives motivates the following definition:

Definition 2.6. A CW-spectrum is a spectrum E in which the spaces En are CW-complexes with a 0-cell as basepoint for all n ∈ N and such that the structure maps are inclusions of subcomplexes. A CW-subspectrum of a CW-spectrum E is a subspectrum A as defined earlier with the requirement that An is a subcomplex of En for all n ∈ N.

This way, for each cell ei of En, different from the basepoint, its suspen- sion becomes a cell of dimension i + 1 in En+1. We say that two cells eα of En and eβ of Em, with n ≤ m are equivalent if eα becomes eβ by suspending (and including) a suitable number of times. We will say that a stable cell of E is an equivalence class under this relation. So, a stable cell consists of cells n+k e of En for all n ≥ n0. The dimension of such a stable cells is defined to be k. This way, if we consider the suspension spectrum of a CW-complex X, the cells of X coincide with the stable cells of the spectrum, together with their dimensions. The homology of a CW-spectrum E can be defined in terms of cellular chains. We consider the cellular chain complexes relative to the basepoint C∗(En; G): the structure maps ΣEi → Ei+1 induce inclusions C∗(En; G) → C∗(En+1; G). If we take the union C∗(E; G) = ∪nC∗(En; G), it can be checked that we get a chain complex with one G summand for every stable th cell of E, as for CW-complexes. Then we define Hi(E; G) to be the i homology group of this chain complex. A CW-spectrum with finitely many stable cells is called finite. If the number of cells in each dimension is finite, then the spectrum is of finite type.

1in the union, we identify, for each space in the composition, its copy in the mapping cylinders of the map from and to it. 2.2 CW-Spectra 6

Remark 2.7. Note that a CW-spectrum can also have cells in negative dimension: consider, for example, the CW-spectrum X defined by 1 2 Xn = S ∨ S ∨ ... for all n ≥ 0, with structure map the obvious inclusion 2 3 Σ(Xn) = S ∨ S ∨ ...,→ Xn+1 and in which all the factors Si are realized as a disc Di with the boundary glued on one point. This spectrum has one cell for each dimension k ∈ Z. The Adams Spectral Sequence deals only with CW-spectra in which the dimension of the stable cells is bounded below. A CW-spectrum with this property is called connective. The objects in the category we are going to build are of course CW- spectra. Instead, it is not that easy to find a suitable class of maps and it will require some work to get to a definition of morphism in our category. A reasonable goal would be to have that a morphism between two CW-spectra induces a homomorphism on homotopy. The most obvious choice would be to take maps that commute with the structure maps of the spectra. Namely, we can give the following definition: 2 Definition 2.8. If E and F are CW-spectra and r ∈ Z, a strict map of degree r from E to F is a sequence of maps f = (fn)n∈Z with fn : En → Fn−r a basepoint-preserving map for all n ∈ Z such that the following diagram commutes for all n ∈ Z: - ΣEn En+1

Σfn fn+1 ? ? - ΣFn−r Fn−r+1. We will just say strict map when the degree is 0. Since we define graded maps, it is more convenient to index the spaces of the spectra over the integers rather than over N. We will see soon that this does not cause any substantial difference. Strict maps can be composed in the natural way and the identity of a CW-spectrum is a strict-map. Moreover, if we take a CW-subspectrum E0 of a CW-spectrum E, the inclusion E0 ,→ E is a strict map. We can however weaken the requirements of the definition and still get maps that induce homomorphisms on homotopy. In fact, if we had the maps fn defined only for n ≥ n0 for some n0, we would still get an induced map π (E) → π (F ), since we can think of π (X) as lim π (E ) and similarly i i−r i −→ i+n n for πi(F ). Actually, we can relax the condition some more. To do this, we first need a definition: 2strict maps are called functions in [Ada74]. 2.2 CW-Spectra 7

Definition 2.9. A CW-subspectrum A of a CW-spectrum E is called cofinal i if, for all n ∈ N and for every cell e of En, there exists a k ∈ N such that k i Σ (e ) is a cell of An+k. Lemma 2.10. If E0, E00 are cofinal subspectra of a CW-spectrum E0, then their intersection is also cofinal in E.

i Proof. Let n ∈ N and let e be a cell of En. By definition, there exist k1 and k such that Σk1 (ei) is a cell of E0 and Σk2 (ei) is a cell of E00 . 2 n+k1 n+k2 k i Assume k1 ≤ k2: if we suspend the cell Σ 1 (e ) for k2 − k1 times, we get that Σk2 (ei) is a cell of E0 . This shows that E0 ∩ E00 is cofinal. n+k2 Let E be a CW-spectrum and F a spectrum. Let E0, E00 be cofinal subspectra of E and f : E0 → F , g : E00 → G strict maps of degree r. We say that f and g are equivalent if there exists a cofinal subspectrum E¯ of E contained in E0 and E00 on which f and g coincide. We leave it as an exercise for the reader to check that this is an equivalence relation.

Definition 2.11. A map (of degree r) E → F of CW-spectra is an equiva- lence class of strict maps (of degree r) defined on cofinal subspectra.

Note that the previous lemma tells us that, if we take a map defined on a cofinal subspectrum E0, we get the same map restricting f to another cofinal subspectrum E00 ⊆ E0. The composition of maps of CW-spectra is defined by taking represen- tatives, choosing them in a way such that their composition is defined. It is well defined by the following Proposition:

Proposition 2.12. (a) Let f : E → F be a strict map of degree r of CW- spectra and F 0 ⊆ F a cofinal subspectrum. Then, there exists a cofinal subspectrum E0 ⊆ E such that f(E0) ⊆ F 0.

(b) If E0 is a cofinal subspectrum of E and E00 a cofinal subspectrum of E0, then E00 is a cofinal subspectrum of E.

Proof. (a) We set E0 to be the subspectrum made by the cells of the spaces 0 i En which are mapped into Fn−r. This is cofinal, in fact, if e is a cell in En, then it is mapped by f to a finite union of cells of Fn−r and 0 0 so, for some k, their k-fold suspension lies in Fn−r+k, since F cofinal. f is a strict map and so it commutes with the structure maps. Then, the Σk(ei) is mapped in F 0 by f and so it is in E0.

(b) This is a straightforward consequence of the definition.

To see that also a map f : E → F of CW-spectra induces a morphism f∗ : πi(E) → πi−r(F ), assume that f is defined on the cells of a cofinal 2.2 CW-Spectra 8

0 i+n subspectrum E of E. If we take a map S → Xn, its image is compact and hence included in a finite subcomplex of Xn. If we suspend the cells of this 0 subcomplex enough times, they will lie in En+k (for some k) and f is defined i+n on them. The same process can be done for a homotopy S × I → Xn. We can now define homotopies between maps of CW-spectra, in a way all similar to what is done for spaces: for a CW-spectrum E, we define the cylinder spectrum E × I by

(En × I) (E × I)n = , e0 × I where e0 denotes the basepoint. Clearly we have Σ(En × I) = Σ(En) × I. The structure maps of the cylinder spectrum are defined in the obvious way. We have two obvious inclusion maps:

i0, i1 : E → E × I, that correspond to the two ends of the cylinder. Definition 2.13. If f, g : E → F are maps of CW-spectra, we say that they are homotopic if there exists a map of CW-spectra H : E × I → F such that f = H ◦ i0, g = H ◦ i1. We leave to the reader to check that this is an equivalence relation and the composition of maps defined earlier induces the composition for homotopy classes, as in the familiar case of spaces. The set of the homotopy classes of the maps E → F of degree r is denoted [E,F ]r We have finally obtained a satisfactory definition for the morphisms in our category: the objects are CW-spectra and the morphisms of degree r are homotopy classes of maps of degree r. We will just say morphism to mean a morphism of degree 0. Note that a morphism f is an isomorphism in this category if it is a homotopy equivalence of spectra. Remark 2.14. If E0 is a cofinal subspectrum of a CW-spectrum E, we already know that the inclusion E0 ,→ E is a strict map and hence a map of CW-spectra. Moreover, if we take the equivalence class of the identity of E0, it is a map of CW-spectra E → E0, by definition, and one can easily check that the composition of these two maps in any order gives identity. This shows that every CW-spectrum is isomorphic to any of its cofinal subspectra. Till now we have always considered spectra indexed over N, but, if 0 E = (En)n∈Z is a CW-spectrum indexed over Z, we can define E by ( E , if n ≥ 0, E0 = n n {∗}, if n < 0.

E0 is a cofinal subspectrum of E and so they are isomorphic: this shows that it does not really matter if we consider spectra indexed over N or over Z. 2.2 CW-Spectra 9

Remark 2.15. We can always define a subspectrum E0 of E by setting

0 En = ΣEn−1 ⊆ En. This is a cofinal subspectrum of E and so, by the previous remark, they are isomorphic. More generally, we can define the suspension ΣF of a spectrum F by setting (ΣF )n = Σ(Fn). Then, the last assertion can be reformulated by saying that any CW-spectrum E is isomorphic to ΣF where F is the spec- −1 trum defined by Fn = En−1. This spectrum F can be denoted by Σ E, so that E = Σ(Σ−1E). This argument shows that any CW-spectrum is isomorphic to the suspension of another CW-spectrum. In other words, the suspension is invertible in our category. We can iterate this construction: for m ∈ N, we define a CW-spectrum G by:

Gn = En−m.

This way we get that E =∼ ΣmG. Recall that the sphere spectrum S0 is the suspension spectrum of the sphere S0. The following Proposition is saying that our choice for morphisms is actually reasonable:

Proposition 2.16. If E is the suspension spectrum for a finite CW-complex K and F is any spectrum, we have that:

[E,F ] = lim[E ,F ]. r −→ n+r n In particular, if K = S0:

0 r πr(E) = [S ,E]r = [S ,E]0.

n+r Proof. If we start with a map (of spaces) f :Σ K → Fn, we can define a map of spectra f¯: E → F of degree r: f¯n+r : En+r → Fn is taken to be f itself and, since we have ΣEn+r = En+r+1, we are forced to define, for all m > 0: m m Σ f En+r+m = Σ (En+r) −−−→ Fn+m. This defines a strict map on the cofinal subspectrum: ( E , if i ≥ n + m, E0 = i i {∗}, if i < n + m.

n+r m+r If f :Σ K → Fn, g :Σ K → Fm are two maps which are equivalent in the direct limit, it means that, for some p there exists a homotopy between the maps: p−n p+r Σ f p−n Σ K −−−−→ Σ Fn → Fp, 2.2 CW-Spectra 10

p−m p+r Σ g p−m Σ K −−−−→ Σ Fm → Fp. This homotopy induces a homotopy between the map of spectra correspond- ing to f and g and so we have a map:

α: lim[E ,F ] → [E,F ] . −→ n+r n r A map of CW-spectra E → F of degree r is a homotopy class of strict maps defined on a cofinal subspectrum of E. Any of these strict maps is equivalent to one defined on a subspectrum which has the form of E0 defined above and so α is surjective. Moreover, if two maps of CW-spectra are homotopic, the homotopy is induced by a homotopy of maps of spaces, therefore α is injective. Therefore α is a bijection.

Remark 2.17. One of the reasons to consider spectra instead of spaces is that, for E,F CW-spectra, the set [E,F ] is always an abelian group. This is because, as we have seen, a CW-spectrum is always isomorphic to the suspension of another CW-spectrum (and hence also to a double suspension), in other words it can always be de-suspended as many times as we want, allowing us to define an abelian sum operation using the extra coordinates given by the suspension, as one does for homotopy groups of spaces.

Proposition 2.18. If E,F are CW-spectra, the suspension map

[E,F ] → [ΣE, ΣF ] is a group isomorphism.

Proof. See [Ada74, Part III, Theorem 3.7] or [Hat04, p. 10].

Another advantage of working with CW-spectra is that many results which hold for CW-complexes also translate in this context. For instance we have cellular approximation of maps, as for CW-complexes. We have also a version of the Whitehead theorem:

Theorem 2.19. A morphism of CW-spectra f : E → F that induces iso- morphisms f∗ : πi(E) → πi(F ) for all i is an isomorphism and so a homotopy equivalence between e and F .

Proof. See [Ada74, Part III, Corollary 3.5].

Another important result translate to the category of CW-spectra: the Hurewicz theorem. It must be noted that it holds only for connective spectra:

Theorem 2.20. Let E be a connective CW-spectrum. It πi(E) = 0 for all i < n, then there exists an isomorphism h: πn(E) → Hn(E; Z). Proof. See [Mar83, Theorem 6.9]. 2.3 Exact sequences for a cofibration of CW-spectra 11

2.3 Exact sequences for a cofibration of CW-spectra In order to talk about cofibrations, we need to define mapping cylinders and mapping cones for a map of CW-spectra. If f : E → F is a map of CW-spectra, we can pick as a representative for it a strict map f 0 : E0 → F defined on a cofinal subspectra E0 of E. Now 0 we can deform fn to be cellular as a map of CW-complexes for each n and define the mapping cylinder of f to be the CW-spectrum Mf with:

0 (Mf )n = Mfn ,

0 where Mfn is a reduced mapping cylinder. This defines a CW-spectrum, 0 0 0 0 since f is a strict map and ΣMfn = MΣfn . If we replace X with a co- final subspectrum, Mf is replaced with a subspectrum cofinal in it, hence isomorphic. Therefore the definition depends uniquely only on f, up to equivalence. The familiar deformation retraction of Mf 0n onto Fn, for each n, give a deformation retraction of the spectrum Mf onto F . The mapping cone, is built up the same way, taking the reduced mapping 0 cones of the maps fn instead of the cylinders. The resulting spectrum is denoted Cf . For an inclusion of a subspectrum A,→ E, the mapping cone can be written as X ∪ CA, as one does for CW-complexes, defining the cone cylinder CA similarly to what we did for the cylinder spectrum. Definition 2.21. A subspectrum A of a CW-spectrum E is called closed i i if, when a cell e of En has an iterated suspension lying in An+k, then e is in An. If A is a closed subspectrum of E, we can form the quotient spectrum E/A setting (E/A)n = En/An. We have a map E ∪ CA → E/A, whose components are the homotopy equivalences of CW-complexes En ∪ CAn → En/An. We can then apply Theorem 2.19 to get a homotopy equivalence X ∪ CA ' X/A. As for spaces, from the inclusion of a subspectrum, we can form a cofi- bration sequence: A,→ E → E ∪ CA, which can we extended to the right by taking the mapping cone of the last inclusion: A,→ E → E ∪ CA → (E ∪ CA) ∪ CE and we have that (E ∪ CA) ∪ CE ' (E ∪ CA)/E ' ΣA, so we can go on extending the sequence with ΣA → ΣE, just as for spaces:

A,→ E → E ∪ CA → ΣA → ΣE → · · · (2.1)

And, as for spaces, we have an associated exact sequence:

[A, F ] ← [E,F ] ← [E/A, F ] ← [ΣA, F ] ← [ΣE,F ] ← · · · (2.2) 2.3 Exact sequences for a cofibration of CW-spectra 12

But with CW-spectra we have also another exact sequence associated to a cofibration, which is not present for cofibrations of spaces. The reason is basically that we can always de-suspend spectra, which is is not true for spaces. Proposition 2.22. For a cofibration A,→ E → E ∪ CA, the following sequence is exact: [F,A] → [F,E] → [F, E/A] → [F, ΣA] → [F, ΣE] → · · · (2.3) Proof. The sequence (2.1) is built including each space in the mapping cone of the map that is entering that space. So, it is enough to prove the exactness of the sequence: [F,A] → [F,E] → [F,E ∪ CA]. It is clear that the composition of the two maps is zero. Suppose now that we have f : F → E that becomes nullhomotopic when including E into E ∪CA. This homotopy F × I → E ∪ CA gives a map H : CF → E ∪ CA (since CF can be seen as the quotient of F × I where we mod out one face of the cylinder) which forms a commutative square with f: F ⊂ - CF

f H ? ? E ⊂ - E ∪ CA. The two rows complete to cofibrations and we get the two next vertical maps at the right of the square to fill the following diagram, which commutes up to homotopy: 1 1 F - F ⊂ - CF - ΣF - ΣF

f H k Σf ? ? ? ? Σi A ⊂ - E ⊂ - E ∪ CA. - ΣA - ΣE. By Proposition 2.18, k = Σl, for some l ∈ [F,A] and, by commutativity, Σf ' Σ(i)Σ(l) = Σ(il). Since the suspension is an isomorphism, this implies f ' il and then f is in the image of [F,A] → [F,E]. This completes the proof of the exactness. This exact sequence we built for a cofibration of CW-spectra is the anal- ogous of the exact sequence [Y,F ] → [Y,E] → [Y,B] for a fibration of spaces F → E → B. In other words, in the category of CW-spectra, cofibrations are the same as fibrations. Given two CW-spectra A and B we can form their wedge sum A ∨ B in the obvious way, setting (A ∨ B)n = An ∨ Bn for all n. The structure maps come from pasting the ones of A and B, since Σ(An ∨ Bn) is homeomorphic to ΣAn ∨ ΣBn. Moreover the inclusions of the two factors, A → A ∨ B ← B are strict maps of spectra. 2.4 Cohomology for CW-spectra 13

Remark 2.23. For cofibration of the pair (A ∨ B,A), the exact sequence of Proposition 2.22:

· · · → [F,A] → [F,A ∨ B] → [F,B] → · · · has a splitting [F,B] → [F,A ∨ B] given by inclusion and so we have that [F,A ∨ B] =∼ [F,A] ⊕ [F,B]. This clearly generalizes to finite wedge sums.

2.4 Cohomology for CW-spectra Definition 2.24. Given a CW-spectrum E, we define the nth E-cohomology of a CW-spectrum X as:

n E (X) = [X,E]−n.

An element f ∈ [X,E]−n is a homotopy class of maps of spectra X → F of degree −n. It corresponds to a strict map f 0 defined on a subspectrum 0 0 X ⊆ X with components fi : Xi−n → Ei. As shown in Remark 2.15, the spectrum X is isomorphic to ΣnY , where Y is the spectrum defined by Yi = Xi−n for all i. Then we have:

[Σ−nX,E] = [Σ−nΣnY,E] = [Y,E], and a map Y → E of degree 0 corresponds to a strict map f 0 defined on a 0 0 cofinal subspectrum Y ⊆ Y with components fi : Yn = Xi−n → En. Thus, En(X) can be also written as [Σ−nX,E]. One check easily that, for all n, En(X) is a contravariant functor from the category of CW-spectra to abelian groups in the variable X. The reason for the name “E-cohomology” is that En satisfies a series of properties analogous to the axioms for a reduced cohomology theory on spaces (see, for instance, [May99, Ch. 19] for a reference on them):

Proposition 2.25. If E is a CW-spectrum, the functor En satisfies the following properties:

(a) For a cofibration A,→ X → X/A, the sequence:

En(X/A) → En(X) → En(A)

is exact.

(b) There is a natural isomorphism En(X) → En+1(ΣX).

(c) For a finite wedge sum of CW-spectra A1 ∨ · · · ∨ Ak, there is an iso- morphism k n ∼ Y n E (A1 ∨ · · · ∨ Ak) = E (Ai). i=1 2.4 Cohomology for CW-spectra 14

(d) A map of CW-spectra f : X → Y that is a weak equivalence induces a isomorphism: f ∗ : En(Y ) → En(X).

Proof. (a) We can just use the exact sequence (2.2).

(b) Writing En(X) as [Σ−nX,E], we have:

n ∼ −n ∼ E (X) = [X,E]−n = [Σ (X),E] = −n−1 ∼ n+1 [Σ (Σ(X)),E] = [Σ(X),E]−n−1 = E (ΣX).

n (c) By definition, E (A1 ∨ · · · ∨ Ak) = [A1 ∨ · · · ∨ Ak,E]−n. To a map f : A1 ∨ · · · ∨ Ak → E we can associate, for 0 ≤ i ≤ k, fi : Ai → E by pre-composing with the inclusion Ai → A1 ∨ · · · ∨ Ak and, conversely, a sequence of maps (fi : Ai → E) define a map from the wedge sum A1 ∨ · · · ∨ Ak to E. (d) If f is a weak equivalence, by Theorem 2.19 it is an isomorphism and so the induced map En(Y ) → En(X) is also an isomorphism.

It is interesting to look at the E-cohomology functors for a CW-spectrum E that we know: given an integer n and an abelian group G, recall that we have the shifted Eilenberg-MacLane spectrum K(G, m), with

K(G, m)n = K(G, m + n).

Note that:3 HGn = K(G, m)n−m. By Remark 2.15, HG is equivalent to Σ−mK(G, m). Hence, by Proposition 2.18, we have a natural isomorphism:

HGm(X) = [Σ−mX,HG] =∼ [X, ΣmHG] = [X,K(G, m)]. (2.4)

It is also possible to define a cohomology theory on CW-spectra in a more direct way, as we did for homology, building the cellular cochain complexes for stable cells and taking the cohomology of the complex, the same way one does for CW-complexes. For example, [Hat04] follows this approach. The resulting cohomology H∗(X; G) is isomorphic to the HG-cohomology ([Hat04, Prop. 2.4]). Moreover, if we apply this functor to the suspension spectrum of a basepointed CW-complex X, it coincides with the ordinary reduced cohomology of X with coefficients in G. If one defines the spectra cohomology with the cellular cochain complex, it is clear that, for a CW-spectrum of finite type, we get finitely generated

3HG is the Eilenberg-MacLane spectrum for the group G, defined in section 2.1. 2.4 Cohomology for CW-spectra 15 cohomology groups. Then, the HG-cohomology groups for a spectrum of finite type are finitely generated. Let us consider the natural transformations of functors:

HGm(−) → HGs(−) for an abelian group G and integers m, s. These natural transformation are cohomology operations and, by (2.4), if we can apply the (contravariant) Yoneda Lemma, we get that the set of them is in bijection with

HGs(K(G, m)) =∼ [K(G, m),K(G, s)] =∼ [HG, K(G, s − m)] =∼ HGs−m(HG)

Namely, a natural transformation HGm(X) → HGs(X) corresponds to the composition: X → K(G, m) → K(G, s). (2.5) So, every natural transformation HG∗(−) → HG∗(−) is identified with an element of HG∗(HG). We have that HG∗(HG) is an abelian group, as in general for any [X,Y ], and it is also a ring, with multiplication given by composing maps. It can be easily checked that, for any CW-spectrum X of finite type, these facts give to HG∗(X) a HG∗(HG)-module structure, with scalar multiplication given again by the composition. Setting G = Z/p for a prime number p, we define the modulo p Steenrod Algebra: ∗ Ap = (HZ/p) (HZ/p), the algebra of stable cohomology operations with Z/p coefficients. It is a Z/p-algebra and it is generated by the Steenrod operations: the Steenrod i k squares Sq , if p = 2, the Steenrod powers P and the Bockstein map βp, for p > 2 ([Ada74, Ch. III.12]). In the construction of the Adams Spectral Sequence we will need an improved version of the isomorphism seen in Remark 2.23, that works for certain infinite wedges of CW-spectra. Note that here we get an isomorphism Q between [X, ∨iKi] and the direct product i[X,Ki], while for finite wedges we have a direct sum.

Proposition 2.26. The natural map Y [X, ∨iK(G, ni)] → [X,K(G, ni)] i

f 7→ (fi),

pj where fj is the composition of f with the projection ∨iK(G, ni) −→ K(G, nj), is an isomorphism if X is a connective CW-spectrum of finite type and ni → ∞ as i → ∞. 2.4 Cohomology for CW-spectra 16

The proof of the Proposition is based on the two following lemmas. Recall that, if X is a connective CW-spectrum of finite type, the dimension of its stable cells is bounded below and X can be written as the increasing union of its k-skeleta Xk. Since X has finite type, each Xk is finite. Lemma 2.27. Let E∗ be the E-cohomology for a fixed CW-spectrum E and X a connective CW-spectrum. Then there is an exact sequence: 0 → lim1 En−1(Xk) → En(X) → lim En(Xk) → 0, ←− ←− where Xk denotes the k-skeleton of X. Recall that lim En(Xk) denotes the inverse limit of the groups G = ←− k En(Xk), which can be seen as the kernel of the homomorphism: Y Y δ : (Gk) → (Gk), δ((gi)) = (hi), i i ∗ ∗ with hj = gj − ij+1(gj+i), where ik is the map induced on cohomology by the inclusion Xk+1 → Xk. Here lim1 is the cokernel of δ. ←− A result analogous to this Lemma, but for CW-complexes instead that for CW-spectra, is proved in [Hat02, Thm. 3F.8]. and the argument can be easily translated to CW-spectra. The second ingredient for the proof of the proposition is the following algebraic lemma, whose proof is omitted from this project: Lemma 2.28 (Mittag-Leffler criterion). If, for each k, the images of G → G are independent of n, for n big enough, then lim1 G = 0. k+n k ←− k Proof (of Proposition 2.26). If X is a finite spectrum, every space Xj has a finite number of cells. Hence, by cellular approximation, all the factors K(G, ni) with ni bigger than the maximum dimension of the cells in the spaces Xj can be omitted from the direct product and from the wedge sum and so we get back to the finite case seen in Remark 2.23. For the general case, we can see the spectrum X as the union of its skeleta Xk, that are finite. n −n If we denote h (X) = [Σ X, ∨iK(G, ni)] is the cohomology associated to ∨iK(G, ni), by the previous lemma we have the exact sequence: 0 → lim1 hn−1(Xk) → hn(X) → lim hn(Xk) → 0, ←− ←− The third term of the spectral sequence lim hn(Xk) is the product we are Q ←− interested in, i[X,K(G, ni)], because it is the inverse limit of the finite product from the previous case. Then it is enough to show that the term lim1 hn−1(Xk) is zero to get the wanted isomorphism. We can apply the ←− Mittag-Leffler criterion: in fact we have that the images of the maps HGi(Xk+n) → HGi(Xk) are independent of n when k + n > i. Then lim1 hn−1(Xk) = 0 and the ←− proof is complete. 17

3 The Adams Spectral Sequence

We begin the section doing the construction of an exact resolution of the cohomology of a CW-spectrum, before moving on to the construction of the Spectral Sequence. The spectral sequence deals with cohomology with coefficients in Z/p, for a prime p. Hence in the following we will consider only n n ∼ this cohomology, with the notation H (X) = (HZ/p) (X) = [X,K(Z, n)].

3.1 Exact resolutions From now on we are restricting our attention to connective CW-spectra of finite type, so that the cohomology groups are finitely generated. The results of the previous section allow us to use CW-spectra in a much similar way to CW-complexes, in the sense that we don’t need anymore to look at the spaces composing a spectra. Instead, when looking at a sequence of spectra, we will use the notation Xn to refer to one of the spectra in the sequence. Let X be a connective CW-spectrum of finite type. Our goal is to build a free resolution of the cohomology H∗(X), seen as a module over the modulo ∗ p Steenrod algebra Ap. The first step is to take αi generators for H (X), n with a finite number of them in each H (X). By (2.4), each αi can be seen as an element of [X,K(Z/p, ni)] and so, by Proposition 2.26, they determine a map X → ∨iK(Z/p, ni).

We set K0 = ∨iK(Z/p, ni), that is a spectrum of finite type, and then, replacing this map with an inclusion via the mapping cylinder construction, we may form the quotient: X1 = K0/X.

X1 is again a connective spectrum of finite type, then we can repeat the construction with X1 in place of X to build K1 and X2 = K1/X1. This yields the diagram:

- - - - X K0 K1 K2 ···

------

X1 X2 X3 3.2 Constructing the spectral sequence 18

and, applying the cohomology functor, we get the diagram: ∗  ∗  ∗  ∗  H (X) H (K0) H (K1) H (K2) ···

   H∗(X ) H∗(X ) H∗(X ) 1 2 3

   0 0 0 0

∗ which gives a free resolution of H (X) as a Ap-module. n −n n −n If we denote X = Σ Xn and K = Σ Kn, we can rewrite the previ- ous diagram in the form: . .

? X2 - K2

? X1 - K1

? X - K0

by using (2.1) for the cofibrations Xs → Ks → Xs+1 we built above and applying the suspension isomorphism of Proposition 2.18. The diagram is called the Adams tower over X.

3.2 Constructing the spectral sequence In the construction of the exact resolution we defined cofibrations:

Xs → Ks → Xs+1.

Then, by Proposition 2.22, we get exact sequences:

t t t t [Σ Y,Xs] → [Σ Y,Ks] → [Σ Y,Xs+1] → [Σ Y, ΣXs] → · · ·

Recall that we have the isomorphisms given by suspension (Proposition 2.18): t ∼ t−1 [Σ Y, ΣXs] = [Σ Y,Xs], 3.2 Constructing the spectral sequence 19

and so the exact sequences above fit together to form a so-called staircase diagram:

- t+1 - t+1 - t+1 - t+1 - t+1 - [Σ Y,Xs] [Σ Y,Ks] [Σ Y,Xs+1] [Σ Y, ΣKs+1] [Σ Y,Xs+2]

? ? ? - t - t - t - t - t - [Σ Y,Xs−1] [Σ Y,Ks−1] [Σ Y,Xs] [Σ Y, ΣKs] [Σ Y,Xs+1]

? ? ? - t−1 - t−1 - t−1 - t−1 - t−1 - [Σ Y,Xs−2] [Σ Y,Ks−2] [Σ Y,Xs−1] [Σ Y, ΣKs−1] [Σ Y,Xs] ? ? ?

Such a diagram determines an exact couple, which is one of the standard ways to construct a spectral sequence (see [HS97, Chapter VIII] for a de- scription of the construction in general).

Definition 3.1. The spectral sequence obtained from the previous staircase diagram is called the modulo p Adams spectral sequence.

To simplify the writing, we introduce the notation:

Y t πt (X) = [Σ Y,X].

0 S0 This way, taking Y equal to S , the sphere spectrum, we have: πt (X) = πt(X). To identify the terms in the first pages of the spectral sequence, we pro- ceed as follows: by construction, each Ks is a wedge of Eilenberg-MacLane ∗ spaces and we know that its cohomology H (Ks) is a free Ap-module. Y t t The elements of πt (Ks) = [Σ Y,Ks] are tuples (αi) of maps αi :Σ Y → n t ∗ K(Z/p, ni) or, in other words, αi ∈ H i (Σ Y ). Since H (Ks) is a free module, the natural map:

t α∗ ∗ ∗ t [Σ Y,Ks] −→ HomAp (H (Ks),H (Σ Y ))

is an isomorphism. If we introduce the notation Homt to indicate the ho- momorphism that lower degree by t, this can be rewritten as:

Y ∼ ∗ ∗ t ∼ t ∗ ∗ πt (Ks) = HomAp (H (Ks),H (Σ Y )) = HomAp (H (Ks),H (Y )). Therefore, the terms of the first page of the spectral sequence have the form:

s,t t ∗ ∗ E1 = HomAp (H (Ks),H (Y ))

and the differential d1 in this page is formed by the maps:

s,t s,t s+1,t d1 : E1 → E1 , 3.3 The main theorem 20

induced by the map Ks → Ks+1, that we defined when constructing the ∗ s,t Y s+1,t Y exact resolution of H (X). In fact E1 = πt (Ks) and E1 = πt (Ks+1), s,t Y Y so we may view d1 as a map πt (Ks) → πt (Ks+1). This means that the E1 page consists of the complexes

0,t t ∗ ∗ di t ∗ ∗ 0 → HomAp (H (K0),H (Y )) −−→ HomAp (H (K1),H (Y )) 1,t di t ∗ ∗ −−→ HomAp (H (K2),H (Y )) → ... (3.1)

The terms of the next page E2 are, by definition:

s,t s,t s−1,t E2 = ker(d1 )/ Im(d1 ), hence, computing them amounts to compute the homology of the complexes ∗ ∗ (3.1) and so, since the groups H (Ks) are an exact resolution of H (X), we get the Ext groups:

Es,t = Exts,t (H∗(X),H∗(Y )). 2 Ap

3.3 The main theorem Our main result deals with the convergence of the Adams Spectral Sequence.

Theorem 3.2. Let X be a connective CW-spectrum of finite type. Then the Adams Spectral Sequence with E2 terms of the form:

Es,t = Exts,t (H∗(X),H∗(Y )) 2 Ap

Y is convergent to π∗ (X) modulo the subgroup given by torsion elements of order prime to p. In other words:

s,t s,t s+1,t+1 (a) The terms E∞ are isomorphic to the successive quotients F /F Y s,t for a filtration of πt−s(X) in which F is the image of the map πt(Xs) → πt−s(X).

s+n,t+n Y (b) ∩nF is the subgroup of πt−s(X) of the torsion elements of order prime to p.

Y Note that the filtration can very well be infinite: this happens if πt−s(X) s,t is infinite, since all the elements Er of the spectral sequence are finite dimensional Z/p vector spaces. In fact the terms on the E1 page are finite t ∗ ∗ dimensional, because they have the form HomAp (H (Ks),H (Y )), and this implies that all the terms in the next pages are finite dimensional as well. Let us introduce the following terminology: a sequence of spectra (Li) with maps Z → L0 → L1 → ... such that the composition of two con- secutive maps is nullhomotopic is called a complex on Z. A complex such 3.3 The main theorem 21

that every spectrum L1 is a wedge of Eilenberg-MacLane spectra is named Eilenberg-MacLane complex. If the sequence induced in cohomology:

∗ ∗ ∗ 0 ← H (Z) ← H (L0) ← H (L1) ← ... is exact, then the complex is called resolution. Let us prove a lemma that will be used in the proof of the theorem:

Lemma 3.3. If we are given a resolution of Z:

Z → L0 → L1 → ..., an Eilenberg-MacLane complex:

X → K0 → K1 → ... and a map f : Z → X, then we can build maps fi forming the following diagram, commutative up to homotopy: - - - - Z L0 L1 L2 ...

f f0 f1 f2 ? ? ? ? - - - - X K0 K1 K2 ... Proof. Since the composition of two consecutive maps is nullhomotopic, we can factor the maps as in the diagram:

L0/Z = Z1 L1/Z1 = Z2 - -

------Z L0 L1 L2 ...

f f0 f1 f2 ? ? ? ? - - - - X K0 K1 K2 ... - -

- ? - ? K0/X = X1 K1/X1 = X2

The map X → K0 = ∨K(Z/p, ni) corresponds to a sequence of αi : X → n K(G, ni) and we can view each αi as a class in H i (X), by (2.4). Since the ∗ ∗ map H (L0) → H (Z) is surjective (because the first row is a resolution), n ∗ there exist, for all i, some βi ∈ H i (L0) such that f (βi) = αi. Each βi gives a homotopy class in [L0,K(G, ni)], again by (2.4), and so we get a map f0 : L0 → K0, forming a homotopy-commutative square. By exactness, ∗ ∗ the map H (L1) → H (Z1) is surjective, so we can repeat the argument to build f1 and further for all the other fi’s. 3.3 The main theorem 22

We can now prove out Theorem about the convergence of the Adams spectral sequence. Proof (of Theorem 3.2). We prove the theorem in the case of the ordinary homotopy groups, in which Y is the sphere spectrum S0: the proof for an arbitrary spectrum Y contains only minor changes. We start from (b) and ∗ for the proof we take Y to be the sphere spectrum, so that H (Y ) = Z/p: since the terms of the E1 page are Z/p vector spaces, as observed, they s,t t ∗ have the form E1 = πt(Ks) = HomA(H (Ks), Z/p). So, by exactness, the vertical maps in the staircase diagram are isomorphisms on the torsion of order prime to p, therefore we have that the torsion of order prime to p is s+n,t+n contained in ∩nF . To prove the other inclusion, we do first a special case: suppose that π∗(X) is entirely of p-torsion. We want to build a Eilenberg-MacLane com- plex over X: X → L0 → L1 → ... as follows: take n such that πn(X) is the first non-trivial homotopy group. Then take generators αi : X → K(Z/p, n) n for H (X) and let L0 be a wedge of Eilenberg-MacLane spectra K(Z/p, n), one for each element αi. This way the map α: X → L0 induces isomorphism on cohomology and also on homology. By the Hurewicz Theorem 2.20, on homotopy it is the map πn(X) → πn(X) ⊗ Zp. Now, if we make the map X → L0 into an inclusion and look at Z1 = L0/X, we have πi(Z1) = 0 for i ≤ n and πn+1(Z1) = ker(πn(X) → πn(L0)), so πn+1(Z1) has a smaller order than πn(X). Then we can repeat the process with Z1 in place of X and build a map Z1 → L1 in the same way, with L1 a wedge of Eilenberg- MacLane spectra. Turning this map into an inclusion, we form Z2 = L1/Z1, whose first non-trivial homotopy group is πn+1(Z2) and it has a smaller or- der than πn+1(Z1). Continuing the process for a finite number of steps, we will get to Zk, with πi(Zk) = 0 for i ≤ n + k. At that point we consider πn+k+1(Zk) and repeat the steps. This way we get the wanted complex i −i X → L0 → L1 → ... . Now, if we denote Z = Σ Zi, we get the Adams tower associated to the complex, with maps · · · → Z2 → Z1 → X. By the previous construction, the map induced by Z1 → X on homotopy is an isomorphism on all degrees except on πn for which it is the inclusion of a proper subgroup. The same for Z2 → Z1, until we reach the step k for which k πn(Z ) = 0 and the inclusion of a proper subgroup is in degree n + 1. The j tower continues: after another finite number of steps, πn+1(Z ) becomes j zero and so on: this implies that, for each i, πi(Z ) is zero if j is big enough. Now we can apply Lemma 3.3, to get a map from the resolution of X that we used to build the spectral sequence and the complex X → L0 → L1 → ... . If we pass to the associated towers and apply homotopy, we get the diagram: - 2 - 1 - ... πi(X ) πi(X ) πi(X)

= ? ? ? - 2 - 1 - ... πi(Z ) πi(Z ) πi(X) 3.3 The main theorem 23

If there was an element in πi(X) pulling back in all the groups in the first row, it would do the same in the second row. But the Zi’s have been constructed j so that πi(Z ) = 0 for j big enough, so this is impossible so the intersection s,t s the groups F = Im(πt(Xs) → πt−s(X)) = Im(πt−s(X ) → πt−s(X) have empty intersection, so (b) is proved in the case of π∗(X) entirely of p-torsion. To prove this inclusion in general, let us consider an element α ∈ πn(X) that is either of infinite order or of p-torsion. We shall prove that it is s+n,t+n not contained in the intersection ∩nF . Since the order of α is either infinite of pt for some t ≥ 0, we can pick a positive integer k such that α is k not equal to p times another any other element of πn(X). If we consider the pk identity map in [X,X] and add it to itself pk times, we get a map X −→ X pk and we can form a cofibration X −→ X → H. By (2.1), we get a long exact sequence in homotopy:

pk · · · → πi(X) −→ πi(X) → πi(H) → ...

k pk in which the map p is x 7→ x . By exactness, we have that πi(H) is entirely of p-torsion for all i. We have a map X → H from the cofibration and so, by the Lemma, we get a map from the tower over X associated with the free resolution to the tower over H, as done in the previous case. By the way we have chosen α and k, we have that α is mapped to a non-zero element by the map πn(X) → πn(H), by looking at the previous exact sequence in j degree n. If our element α pulled back in all the groups πi(X ) in the tower over X, it would do the same in the tower over Z, but this is impossible by the case we proved earlier, because π∗(Z) is entirely of p-torsion. Hence α s+n,t+n is not contained in the intersection ∩nF and so (b) is proved. For (a), we look at the rth derived couple associated to the staircase diagram. We have the diagram:

Er s,t kr- - Er Ar - Ar ir ? h Ar ? Ar

We shall prove that, if r is big enough, then ir is injective: injectivity on elements of infinite order or on torsion elements of order prime to p come from the exactness, since the in the E columns of the staircase diagram we have Z/p vector spaces. For elements of p-torsion, we know from (b) that s,t Ar does not contain such elements if r is sufficiently big, therefore the claim is proved. 3.3 The main theorem 24

This implies that, for r big enough, the map preceding ir in the exact s,t couple, kr, is zero, hence the differential dr at Er is also zero. For the s,t differential dr mapping to Er , we have that is is also zero if r is sufficiently big, because it originates at a zero group, since the terms in the E column s,t s,t of the staircase diagram are zero under some level. Hence Er = Er+1 from a certain value of r, since the differentials become zero. Since kr is zero for s,t r big enough, then Er is the cokernel of the map h in the diagram, which s+1,t+1 s,t s,t s,t s+1,t+1 is the inclusion F → F and so E∞ =∼ F /F . We end this section with a result that will be useful for the calculations: if we want to use the spectral sequence with the spectrum Y equal to the sphere spectrum S0, as in the case of the computation of stable homotopy groups, we have a way to simplify the calculation of Exts,t (H∗(X), ): it Ap Zp is to build a minimal free resolution for H∗(X). A free resolution

∗ 0 ← H (X) ← M0 ← M1 ← M2 ← ... is called minimal if during the construction of the free modules, we take the minimal number of generators for each Mi in each degree. The following proposition, which can be proved via an algebraic argu- ment, shows why it is convenient to build such resolutions:

∗ Proposition 3.4. If a free resolution 0 ← H (X) ← M0 ← M1 ← ... is minimal, then all the boundary maps in the complex:

t t t · · · ← HomA(M2, Z/p) ← HomA(M1, Z/p) ← HomA(M0, Z/p) ← 0

s,t ∗ ∼ t are zero and so ExtA (H (X), Z/p) = HomA(Ms, Z/p). Proof. See [Hat04, Ch.2, Lemma 2.8]. 25

4 Calculations

4.1 Stable homotopy groups of spheres One of the most classical applications of the Adams Spectral Sequence is the computation of the stable homotopy groups of spheres. If we take both X and Y equal to the sphere spectrum S0 in Theorem 3.2, we obtain in- formation about the p-component of the stable homotopy groups of spheres s 0 s pπ∗(S ) = pπ∗. In the calculation we will fix p = 2 and denote with A the modulo 2 Steenrod Algebra. The following diagram, that presents a portion of the page E2, is an example of the result of these calculations.

10 9 8 7 6 5 4 3 2 1 h0 h1 h2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13

The vertical axis of the diagram is the coordinate s and the horizontal one t−s, this way each column gives information about one homotopy group: s namely, the column t − s contains information about πt−s. Every dot in the diagram represents one factor Z/2 in E2. s,t s,t s+r,t+r−1 The differential dr in the page Er goes from Er to Er . Hence, in our diagram, dr goes one cell to the left and r cells upward. It can be shown that the vertical lines in the diagram represent multi- s plication by p = 2 in π∗, therefore, for instance, the infinite line of dots in s the column t − s = 0 corresponds to the fact that π0 = Z and so one can keep multiplying by 2 without getting a zero. As a consequence of Theo- n n+1 rem 3.2, each dot in this column is a quotient 2 Z/2 Z for the filtration n+1 n n−1 · · · ⊆ 2 Z ⊆ 2 Z ⊆ 2 Z ⊆ · · · of Z. All the differentials in the portion of E2 drawn here are zero, hence all the dots shown here stay till the spectral sequence converges. To see this, one can use the fact that they are derivations and so satisfy the relation:

dr(xy) = xdr(y) + dr(x)y. 4.1 Stable homotopy groups of spheres 26

This is not obvious, but it is true in our hypothesis as a consequence of [Rav03, Theorem 2.3.3]. The multiplication operation in every page of the spectral sequence corresponds to the graded multiplication that is present s inside the graded stable homotopy group π∗:

s s s πi × πj → πi+j. induced by the compositions Si+j+k → Sj+k → Sk (see Proposition 1.3). The vertical solid lines in the diagram represent multiplication by h0. Let us show how one can prove that the differential dr (in the page Er) originating at (1, 1) is zero. This differential maps to the cell (0, r + 1) and so, if we suppose that it is non-zero, it maps h1 to the non-zero element at r+1 (0, r + 1): we would have dr(h1) = h0 . Then, by the derivation rule:

r+1 r+2 dr(h0h1) = h0dr(h1) + dr(h0)h1 = h0h0 = h0 ,

r+2 and h0 is non-zero, because multiplication by h0 in the column t−s = 0 is an isomorphism. But, looking at the diagram, we see that the product h0h1 is zero, and so the left hand side of the equality is zero, giving a contradiction. Therefore, we must have dr(h1) = 0. To get the picture of the E2 page that we have described, we need to calculate the ExtA(Z/2, Z/2) terms and for this we may work out a minimal free resolution · · · → M2 → M1 → M0 → Z/2 → 0 of Z/2 as a A-module. The process begins by taking as M0 a copy of A, with a generator A0 in degree zero that maps to the generator of Z/2, as shown in the diagram of the next page. For the resolution to be exact, we need to choose M1 so that it maps onto the kernel of the map M0 → Z/2, I I that is composed by all the terms of the form Sq A0, where Sq , for I = i1 in (i1, i2, . . . , in), denotes the product of Steenrod squares Sq ... Sq . So we 1 1 need a generator B1 in M1 that maps to Sq A0. This implies that Sq B1 1 1 maps to Sq Sq A0 = 0, by the Adem relations. Hence we need a new 2 generator B2 in the position s = 1, t − s = 1, mapping to Sq A0. The process continues like this: note that in the diagram, in the column s, we have the elements of degree k in the row t − s = k − s. We use coordinates s and t − s on the axes of the diagram to use the same notation that is used in the spectral sequence. For this reason, the maps of the resolution, which have degree 0, are sloping downward to the left. These maps are displayed as arrows in the chart and are omitted for elements that map to 0. An arrow with more than one head means that the element from which the arrow originates maps to the sum of the elements pointed. All the resolution can be computed inductively, proceeding row by row and from left to right in each row: in every entry (s, t − s) of the diagram we compute the boundary map on the elements that are already present 4.1 Stable homotopy groups of spheres 27

s F0 F1 F2 F3 F4 F5 t−s

0 A0 B1 C2 D3 E4 F5

1 1 1 1 1 1 1 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5

B2

2 2 2 2 2 2 2 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5

1 Sq B2 C4

2,1 2,1 2,1 2,1 2,1 2,1 3 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5

3 3 3 3 3 3 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5

2 1 Sq B2 Sq C4

B4 C5 D6

3,1 3,1 3,1 3,1 3,1 3,1 4 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5

4 3 4 4 4 4 Sq A0 Sq B2 Sq C2 Sq D3 Sq E4 Sq F5

4 2 1 Sq B1 Sq C4 Sq D6 2,1 1 Sq B2 Sq C5 1 Sq B4

4,1 4,1 4,1 4,1 4,1 4,1 5 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5 5 Sq A0 4 5 5 5 5 Sq B2 Sq C2 Sq D5 Sq E4 Sq F5 3,1 3 2 Sq B2 Sq C4 Sq D6 5 2 Sq B1 Sq C5 2 2,1 Sq B4 Sq C4

4,2 4,2 5,1 5,1 5,1 5,1 6 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5 5,1 Sq A0 5,1 4,2 4,2 4,2 4,2 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5 6 Sq A0 6 6 6 6 6 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5 2,1 2,1 2,1 Sq B4 Sq C5 Sq D6 4,1 3,1 3 Sq B2 Sq C4 Sq D6 5 4 Sq B2 Sq C4 Sq3 B Sq3 C . 4 5 . C8

4,2,1 4,2,1 4,2,1 4,2,1 4,2,1 7 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 6,1 6,1 6,1 Sq A0 Sq B1 Sq C2 6,1 6,1 Sq D3 Sq E4 5,2 5,1 4,1 Sq A0 Sq B2 Sq C4 7 5,2 5 Sq A0 Sq B1 Sq C4 4,2 7 Sq B2 Sq C2 . . . 7 3,1 . . . Sq B1 Sq C5 3,1 Sq B4 . 6 . Sq B2 . 4 Sq B4 . B8 . C9 D10 E11 4.2 The homotopy of the spectrum ko 28

(and have the form SqI times a generator introduced in a previous row, for all the admissible monomials SqI of the appropriate degree). Then, to decide if we need a new generator in the cell, we look at the map going from (s − 1, t − s + 1) to (s − 2, t − s + 2): if it is not injective, then we have to introduce a new generator. It does not matter if in the next row we will introduce new generators in (s − 1, t − s + 1), because generators map injectively, so they are never in a kernel. This process makes use of the Adem relations, to write the image of the product SqI X as admissible monomials. This example gives the feeling of how one uses this spectral sequence for calculations: first there is the algebraic problem of building an exact resolution for H∗(X) and computing the homology of the Hom complex of s,t ∗ ∗ the resolution, to get the terms ExtA (H (X),H (Y )). Then one has to understand what differentials dr are non-trivial and to compute them to understand the convergence of the spectral sequence. As shown, E∞ only contains the successive quotients in filtrations of the homotopy groups, so one has still to solve the group extension problem to recover the homotopy groups.

4.2 The homotopy of the spectrum ko

In this section we will show a computation of the 2-component 2π∗(ko) of the homotopy of the spectrum ko, the spectrum of real connective K-theory. In this specific case, the computation of an exact resolution of H∗(ko) is easier, because, via a certain change-of-ring isomorphism, we can avoid working with the (mod 2) Steenrod algebra A and use instead the finite-dimensional subalgebra A(1), which is defined as the subalgebra generated by Sq1 and 2 Sq . This is a module (or, in other words, a vector space) over Z/2. By an explicit computation of the products of elements of A(1), applying the Adem relations, one sees that the following elements give an explicit basis for it, so A(1) is 8-dimensional over Z/2:

B = {Sq0, Sq1, Sq2, Sq3, Sq2 Sq1, Sq3 Sq1, Sq2 Sq1 Sq2, Sq5 Sq1}.

We recall a shortened notation that is used to denote products of Steenrod squares: Sqi1,..., n = Sqi1 ... Sqin . In the following computations we will use repeatedly the Adem relations, so we list here the ones that we are going to 4.2 The homotopy of the spectrum ko 29 need:

Sq0 = 1 Sq1,1 = 0 Sq1,2 = Sq3 Sq1,3 = 0 Sq2,2 = Sq1,2,1 = Sq3,1 Sq2,1,2 = Sq2,3 Sq2,2,2 = Sq3,1,2 = Sq5,1 Sq2,1,2,1 = Sq5,1 Sq1,2,2 = Sq3,2 = 0 Sq2,2,1 = 0.

All elements in our basis for A(1) have degree lower or equal than 6, hence I every product of squares Sq where I = (i0, . . . , in) with i0 + ··· + in > 6 is automatically zero. As said, our aim is to compute the Adams Spectral Sequence for p = 2 to compute 2π∗(ko). We will sketch here the reason for which we can switch to A(1) in this case. For a more detailed explanation of this, we refer the reader to [Rav03, Ch. 3.1]. s,t By Theorem 3.2, the terms E2 of the spectral sequence converging to the 2-component of π∗(ko) are:

s,t s,t ∗ E2 = ExtA (H (ko), Z/2). Now, we have that H∗(ko) is a module over A(1) and, by [Ada74, Part III, Prop. 16.6]: ∗ H (ko) = A//A(1) = A ⊗A(1) Z/2.

So we can substitute in the equation for the terms of E2 and apply a change- of-ring isomorphism, obtaining:

s,t s,t s,t E2 = ExtA (A ⊗A(1) Z/2, Z/2) = ExtA(1)(Z/2, Z/2).

Hence we can compute a free resolution of Z/2 as a A(1)-module to compute this ExtA(1). To do this we proceed as we did in the previous section for computing a minimal free resolution of Z/2 over A, but in this case the computation is easier, since A(1) is finite dimensional. We want to construct free A(1)-modules Mi such that

· · · → M2 → M1 → M0 → Z/2 → 0 is exact. Our computation is displayed in the chart in the following page. Note that the chart is indexed in the same way of the spectral sequence, with s on horizontal axis and t − s on the vertical one. This choice is convenient to translate the information to the spectral sequence, but we have to be careful because this introduces a vertical shifting in the columns: the module Mi, which is in the column i, has its part of degree k in the row k−s. For example, the entry (3, 1) contains the 4-degree part of M3. Another consequence is 4.2 The homotopy of the spectrum ko 30

s F0 F1 F2 F3 F4 F5 F6 F7 t−s 0 A0 B1 C2 D3 E4 F5 G6 H7

1 1 1 1 1 1 1 1 1 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5 Sq G6 Sq H7

B2

2 2 2 2 2 2 2 2 2 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5 Sq G6 Sq H7 1 Sq B2 C4

2,1 2,1 2,1 2,1 2,1 2,1 2,1 2,1 3 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5 Sq G6 Sq H7 3 3 3 3 3 3 3 3 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5 Sq G6 Sq H7 2 1 Sq B2 Sq C4

3,1 3,1 3,1 3,1 3,1 3,1 3,1 3,1 4 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5 Sq G6 Sq H7 2,1 2 Sq B2 Sq C4 D7 E8 F9 G10 H11 3 Sq B2

2,1,2 2,1,2 2,1,2 2,1,2 2,1,2 2,1,2 2,1,2 2,1,2 5 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5 Sq G6 Sq H7 3,1 3 1 1 1 1 1 Sq B2 Sq C4 Sq D7 Sq E8 Sq F9 Sq G10 Sq H11 2,1 Sq C4

5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 6 Sq A0 Sq B1 Sq C2 Sq D3 Sq E4 Sq F5 Sq G6 Sq H7 2,1,2 3,1 2 2 2 2 2 Sq B2 Sq C4 Sq D7 Sq E8 Sq F9 Sq G10 Sq H11

5,1 2,1,2 3 3 3 3 3 7 Sq B2 Sq C4 Sq D7 Sq E8 Sq F9 Sq G10 Sq H11 2,1 2,1 2,1 2,1 2,1 Sq D7 Sq E8 Sq F9 Sq G10 Sq H11

5,1 3,1 3,1 3,1 3,1 3,1 8 Sq C4 Sq D7 Sq E8 Sq F9 Sq G10 Sq H11 E12 F13 G14 H15

2,1,2 2,1,2 2,1,2 2,1,2 2,1,2 9 Sq D7 Sq E8 Sq F9 Sq G10 Sq H11 1 1 1 1 Sq E12 Sq F13 Sq G14 Sq H15 F14

5,1 5,1 5,1 5,1 5,1 10 Sq D7 Sq E8 Sq F9 Sq G10 Sq H11 2 2 2 2 Sq E12 Sq F13 Sq G14 Sq H15 1 Sq F14 G16

2,1 2,1 2,1 2,1 11 Sq E12 Sq F13 Sq G14 Sq H15 3 3 3 3 Sq E12 Sq F13 Sq G14 Sq H15 2 1 Sq F14 Sq G16

3,1 3,1 3,1 3,1 12 Sq E12 Sq F13 Sq G14 Sq H15 2,1 2 Sq F14 Sq G16 H19 3 Sq F14

2,1,2 2,1,2 2,1,2 2,1,2 13 Sq E12 Sq F13 Sq G14 Sq H15 3,1 3 . Sq F14 Sq G16 . 2,1 Sq G16

5,1 5,1 5,1 5,1 14 Sq E12 Sq F13 Sq G14 Sq H15 2,1,2 3,1 Sq F14 Sq G16

5,1 2,.1,2 15 Sq F14 Sq . G16 4.2 The homotopy of the spectrum ko 31 that the maps of the resolution, which have degree 0, are sloping downward to the left. These maps are displayed as arrows in the chart and are omitted for elements that map to 0. An arrow with two heads means that the element from which the arrow originates maps to the sum of the two elements. The first step of the resolution is of course to take as M0 a copy of A(1) ¯ with a generator A0 of degree 0 which is mapped to the generator 1 ∈ Z/2. The generator A0 originates the 8 elements of the first column, given by products with the elements of the basis B of A(1). In general, every time that we introduce a generator in the cell (s, t − s), we get 8 elements that reside in the entries from (s, t − s) to (s, t − s + 6). In the chart, we write generators in red. I The kernel of the map M0 → Z/2 consists of the products Sq A0 for I 0 Sq ∈ B except for Sq = 1. Then, we have to construct M1 so that it maps on this kernel: to start with, we need a generator B1 of degree 1 mapping to 1 i Sq A0 in the first column. This defines the map on all the elements Sq B1: 1 1 1 2 for instance, we have that Sq B1 maps to Sq Sq A0 = 0 and Sq B1 maps 2,1 2 to Sq A0. Now one notes that no element is mapping to Sq A0, therefore 2 we need a generator of M1 mapping to it. Since Sq A0 has degree 2, so does the generator, which we denote with B2. In general we respect the convention that the subscript of a generator indicates its degree. Now we can compute the map on the elements of M1 originating from B2 and we see that we do not need any other generator in this column, because we are already hitting the entire kernel of M0 → Z/2. Then we can continue the process to compute every column of the chart: since we want to obtain a minimal resolution, we introduce a new generator only when it is needed to map on the kernel of the previous map. After few columns, some patterns emerge: namely, we have a generator in every entry of the row t − s = 0, while there is only the generator B2 for 3 2 s = 1 and only C4 mapping to Sq B1 + Sq B2. One can proceed row by row in the calculation, going from left to right in the row. In every cell (s, t − s), to decide if a new generator is needed, we have to look at the map from (s − 1, t − s + 1) to (s − 2, t − s + 2): if it is not injective, then we have to introduce a new generator. As said, it does not matter if in the next row we have to introduce new generators in (s − 1, t − s + 1), because generators map injectively, so they are never in a kernel. We see that we need a generator D7 in s = 3, t − s = 4, mapping to the 2,1,2 3 2,1,2 sum Sq C2 + Sq C4. This causes the existence of the elements Sq D3 1 5,1 and Sq D7 in the cell below, that both map to Sq C2. Therefore we have to introduce a new generator in s = 4, t − s = 4, called E8, to kill these two elements and so on for all the row t − s = 4: hence this row presents an infinite tower beginning in s = 3. We can see inductively for the following three rows that we do not need any new generator. When we come to s = 4, t − s = 8 we have to introduce a new generator E12 and, for a similar 4.2 The homotopy of the spectrum ko 32 argument as before, we get a whole new tower on this row. Now, when we come to the row below, t−s = 9, we see that we are in the same situation that is present in s = 1, t − s = 1: the generators of the row t−s = 8 play the role of there was of the ones in the row t−s = 0: F14 maps to Sq2 times the first generator in the tower. Hence the situation repeats and we get the same pattern of the rows t−s = 0, 1, 2 in the rows t−s = 8, 9, 10, shifted of 4 columns and 8 rows. Similarly, in s = 7, t − s = 12, we have the generator H19 that corresponds to D7, shifted by the same number of rows and columns and so again the situation repeats. Hence the resolution is composed by this pattern that repeats periodi- cally, every time shifted of 4 columns to the right and 8 rows up. By Proposition 3.4, since our resolution is minimal, we have the following pattern on the E2 page of the (mod 2) spectral sequence for 2π∗(ko):

10

9

8

7

6

5

4

3

2 h 1 0 h1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13

Also in this diagram, the vertical lines indicate multiplication by h0. The only differential that could be non-trivial is the one going from the column 8k + 1 to the column 8k, for k ∈ N. It is possible to see that it is zero, using again the fact that, in our hypothesis, differentials satisfy the derivation relation dr(xy) = xdr(y) + dr(x)y ([Rav03, Theorem 2.3.3]), as we have done for the case of the homotopy groups of spheres. To see that dr(h1) = 0, we can apply the derivation rule, which gives, since h0h1 = 0:

0 = dr(h0h1) = h0dr(h1) + dr(h0)h1 = h0dr(h1), and so dr(h1) = 0, because multiplication by h0 is an isomorphism between consecutive groups in the column t − s = 0. Hence, the diagram is also a picture of the E∞ page. As in the case of spheres, the vertical lines in the diagram represent th multiplication by 2. In the E∞ page, each dot in the i column represents a 4.2 The homotopy of the spectrum ko 33

successive quotient isomorphic to Z/2 in a filtration of 2πi(ko). The infinite towers of connected dots in the columns t − s = 4k correspond to the fact that one can keep multiplying by 2 without getting zero: since the homotopy groups of a spectrum of finite type are finitely generated, one can prove that these homotopy groups are isomorphic to Z. For t − s = 8k + 1 or t − s = 8k + 2, we have only one quotient Z/2 in the filtration of the homotopy group, which is then isomorphic to Z/2. The absence of dots in the other columns implies that the corresponding homotopy groups are trivial. In conclusion, we get, for n ≥ 0:  , if n ≡ 0 (mod 4), Z 2πn(ko) = Z/2, if n ≡ 1 or 2 (mod 8), 0, otherwise. REFERENCES 34

References

[Ada74] J.F. Adams. Stable homotopy and generalised homology. University of Chicago Press, 1974.

[Hat02] A. Hatcher. Algebraic topology. Cambridge University Press, 2002.

[Hat04] A. Hatcher. Spectral sequences in algebraic topology. Preprint, 2004.

[HS97] P.J. Hilton and U. Stammbach. A course in homological algebra. Springer, 1997.

[Mar83] H.R. Margolis. Spectra and the Steenrod algebra. North-Holland, 1983.

[May99] J.P. May. A concise course in algebraic topology. University of Chicago Press, 1999.

[Rav03] D.C. Ravenel. Complex cobordism and stable homotopy groups of spheres. American Mathematical Society, 2003.